STAMFORD, Conn. — A former trader for Rochdale Securities in Stamford turned himself in Tuesday to the FBI in a fraud scheme relating to a large purchase of stock in Apple Inc., a press release from U.S. Attorney David B. Fein’s office said.

David Miller, 40, of Rockville Centre, N.Y., is accused of scheming to defraud Rochdale Securities by executing a trade to buy 1.625 million shares of Apple on behalf of a client on Oct. 25, the day the company was announcing its earnings for the quarter, the release said. If the stock price had risen, the release said, Miller would have profited.

When the price declined, the client said it had ordered only 1,625 shares, and Miller is accused of falsely claiming that he made a mistake in ordering. Rochdale suffered about $5 million in losses selling the rest of the shares, the release said.

As part of the scheme, Miller is accused of persuading another broker-dealer to take a short position in Apple by selling 500,000 shares, and falsely claiming that he was trading for the account of another company with which he actually had no relationship, the release said. This was an effort to hedge against his purchase of Apple shares, but it put the other broker-dealer in a position for substantial losses, the release said.

Miller was charged with wire fraud and was released on $300,000 bond, the release said. If found guilty he could face up to 20 years in prison, the release said.